AND DOUBLE THETA-FUNCTIONS. 
903 
and the other terms in question thus reduce themselves to 
— i(«, h, bjx, yf —\rri(x.u+y.-\-.y.v+§) t 
which are independent of m, n, and they thus affect each term of the series with the 
same exponential factor. The result is 
S (t !l )( u +~ i (ax+hy), v+~(hx+by) 
= exp {— i(a, h, bjx, yf—\m{x.u-\-y.-\-.y.v ^-’ A(u, v); 
0 
or (what is the same thing) for a, 0, writing a-fx, 0-\-y respectively, we have 
■»(“' ^+b(c l x+hy),v+~(hx + by) 
=exp { — |(a, li, bjx, y)'~ini(x.u+y.+.y.v+S) }.s(^ + ’ j v). 
Taking x, y even, or writing 2x, 2 y for x, y, then on the right hand side we have 
d(^ + _ ’ v), which is=d^ a ’ ^(w, v), 
but there is still the exponential factor. 
11. The formulae show that the effect of the change u, v into u-\-~(ax-\-hy), 
v-\-—{Jix-\-by), where x, y are integers, is to interchange the functions, affecting them 
however with an exponential factor; and we hence say that \(a, li), —.(h, b) are con¬ 
joint quarter quasi-periods. 
The product-theorem. 
12. We multiply two theta-functions 
■&(“’ g ^(u+u, v+v% g/ ){u — U, v—v')] 
it is found that the result is a sum of four products 
/*<.+/)+**W+£)+j \ (2 2v) _ JH—«’)+P.iW-p + i W 2vl 
\ 7 + 7 . o+o / V V — V . 0 — 6 
7“7 
